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ON THE LEAST COMMON MULTIPLE OF Q-BINOMIAL COEFFICIENTS
Victor J. W. Guo1
Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
jwguo@math.ecnu.edu.cn
Received: , Revised: , Accepted: , Published:
Abstract
We first prove the following identity
÷ ¸ · ¸
· ¸!
n
n
n
lcm([1]q , [2]q , . . . , [n + 1]q )
lcm
,
,...,
=
,
0 q 1 q
n q
[n + 1]q
£ ¤
n
where nk q denotes the q-binomial coefficient and [n]q = 1−q
. Then we show that this identity
1−q
is indeed a q-analogue of that of Farhi [Amer. Math. Monthly 116 (2009), 836–839].
1. Introduction
An equivalent form of the prime number theorem states that log lcm(1, 2, . . . , n) ∼ n as
n → ∞ (see, for example, [4]). Nair [7] gave a nice proof for the well-known estimate
lcm{1, 2, . . . , n} ≥ 2n−1 , while Hanson [3] already obtained lcm{1, 2, . . . , n} ≤ 3n . Recently,
Farhi [1] established the following interesting result.
Theorem 1 (Farhi) For any positive integer n, there holds
µµ ¶ µ ¶
µ ¶¶
n
n
n
lcm(1, 2, . . . , n + 1)
lcm
,
,...,
=
.
0
1
n
n+1
(1)
As an application, Farhi shows that the inequality lcm{1, 2, . . . , n} ≥ 2n−1 follows immediately from (1).
1
This work was partially supported by Shanghai Educational Development Foundation under the Chenguang Project (#2007CG29), Shanghai Rising-Star Program (#09QA1401700), Shanghai Leading Academic
Discipline Project (#B407), and the National Science Foundation of China (#10801054).
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2
The purpose of this note is to give a q-analogue of (1) by using cyclotomic polynomials.
n
Recall that a natural q-analogue of the nonnegative integer n is given by [n]q = 1−q
. The
1−q
£M ¤
Qn
corresponding q-factorial is [n]q ! = k=1 [k]q and the q-binomial coefficient N q is defined as

[M ]q !
· ¸

, if 0 ≤ N ≤ M ,
M
= [N ]q ![M − N ]q !
N q 
0,
otherwise.
Let lcm also denote the least common multiple of a sequence of polynomials in Z[q]. Our
main results can be stated as follows:
Theorem 2 For any positive integer n, there holds
÷ ¸ · ¸
· ¸!
n
n
n
lcm([1]q , [2]q , . . . , [n + 1]q )
lcm
,
,...,
=
.
0 q 1 q
[n + 1]q
n q
Theorem 3 The identity (2) is a q-analogue of Farhi’s identity (1), i.e.,
÷ ¸ · ¸
· ¸!
µµ ¶ µ ¶
µ ¶¶
n
n
n
n
n
n
lim lcm
,
,...,
= lcm
,
,...,
,
q→1
0 q 1 q
n q
0
1
n
(2)
(3)
and
lcm(1, 2, . . . , n + 1)
lcm([1]q , [2]q , . . . , [n + 1]q )
=
.
q→1
[n + 1]q
n+1
lim
Although it is clear that
(4)
· ¸
µ ¶
n
n
lim
=
,
q→1 k
k
q
the identities (3) and (4) are not trivial. For example, we have
µ
¶
¡
¢
2
2
4 = lim lcm 1 + q, 1 + q 6= lcm lim(1 + q), lim(1 + q ) = 2.
q→1
q→1
q→1
2. Proof of Theorem 2
Let Φn (x) be the n-th cyclotomic polynomial. The following easily proved result can be
found in [5, (10)] and [2].
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx
Lemma 4 The q-binomial coefficient
£n¤
k q
3
can be factorized into
· ¸
Y
n
Φd (q),
=
k q
d
where the product is over all positive integers d ≤ n such that bk/dc + b(n − k)/dc < bn/dc.
Lemma 5 Let n and d be two positive integers with n ≥ d. Then there exists at least one
positive integer k such that
bk/dc + b(n − k)/dc < bn/dc
(5)
if and only if d does not divide n + 1.
Proof. Suppose that (5) holds for some positive integer k. Let
k≡a
(mod d),
(n − k) ≡ b
(mod d)
for some 1 ≤ a, b ≤ d − 1. Then n ≡ a + b (mod d) and d ≤ a + b ≤ 2d − 2. Namely,
n + 1 ≡ a + b + 1 6≡ 0 (mod d). Conversely, suppose that n + 1 ≡ c (mod d) for some
1 ≤ c ≤ d − 1. Then k = c satisfies (5). This completes the proof.
2
Proof of Theorem 2. By Lemma 4, we have
÷ ¸ · ¸
· ¸! Y
n
n
n
Φd (q),
lcm
=
,
,...,
0 q 1 q
n q
d
(6)
where the product is over all positive integers d ≤ n such that for some k (1 ≤ k ≤ n) there
holds bk/dc + b(n − k)/dc < bn/dc. On the other hand, since
[k]q =
qk − 1
=
q−1
Y
Φd (q),
d|k, d>1
we have
lcm([1]q , [2]q , . . . , [n + 1]q )
=
[n + 1]q
Y
Φd (q).
(7)
d≤n, d-(n+1)
By Lemma 5, one sees that the right-hand sides of (6) and (7) are equal. This proves the
theorem.
2
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4
3. Proof of Theorem 3
We need the following property.
Lemma 6 For any positive integer n, there holds
(
p, if n = pr is a prime power,
Φn (1) =
1, otherwise.
Proof. See for example [6, p. 160].
2
In view of (6), we have
÷ ¸ · ¸
· ¸! Y
n
n
n
Φd (1),
=
lim lcm
,
,...,
q→1
0 q 1 q
n q
d
(8)
where the product is over all positive integers d ≤ n such that for some k (1 ≤ k ≤ n) there
holds bk/dc + b(n − k)/dc < bn/dc. By Lemma 6, the right-hand side of (8) can be written
as
Y
P∞
r
r
r
p r=1 max0≤k≤n {bn/p c−bk/p c−b(n−k)/p c} .
(9)
primes p ≤ n
We now claim that
∞
X
r=1
max {bn/pr c − bk/pr c − b(n − k)/pr c}
0≤k≤n
= max
0≤k≤n
∞
X
(bn/pr c − bk/pr c − b(n − k)/pr c) .
(10)
r=1
P
i
Let n = M
i=0 ai p , where 0 ≤ a0 , a1 , . . . , aM ≤ p − 1 and aM 6= 0. By Lemma 5, the left-hand
side of (10) (denoted LHS(10)) is equal to the number of r’s such that pr ≤ n and pr - n + 1.
It follows that
(
0,
if n = pM +1 − 1,
LHS(10) =
M − min{i : ai 6= p − 1}, otherwise.
It is clear that the right-hand side of (10) is less than or equal to LHS(10). If n = pM +1 − 1,
then both sides of (10) are equal to 0. Assume that n 6= pM +1 −1 and i0 = min{i : ai 6= p−1}.
Taking k = pM − 1, we have
(
0, if r = 1, . . . , i0 ,
bn/pr c − bk/pr c − b(n − k)/pr c =
1, if r = i0 + 1, . . . , M ,
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx
and so
∞
X
5
bn/pr c − bk/pr c − b(n − k)/pr c = M − i0 .
r=1
Thus (10) holds. Namely, the expression (9) is equal to
Y
p
max0≤k≤n
P∞
r
r
r
r=1 (bn/p c−bk/p c−b(n−k)/p c)
primes p ≤ n
µµ ¶ µ ¶
µ ¶¶
n
n
n
= lcm
,
,...,
.
0
1
n
This proves (3). To prove (4), we apply (7) to get
lcm([1]q , [2]q , . . . , [n + 1]q )
=
q→1
[n + 1]q
Y
lim
Φd (1),
d≤n, d-(n+1)
which, by Lemma 6, is clearly equal to
lcm(1, 2, . . . , n + 1)
.
n+1
Finally, we mention that (10) has the following interesting conclusion.
Corollary 7 Let p be a prime number and let k1 , k2 , . . . , km ≤ n, r1 < r2 < · · · < rm be
positive integers such that
bn/pri c − bki /pri c − b(n − ki )/pri c = 1 f or i = 1, 2, . . . , m.
Then there exists a positive integer k ≤ n such that
bn/pri c − bk/pri c − b(n − k)/pri c = 1 f or i = 1, 2, . . . , m.
References
[1] B. Farhi, An identity involving the least common multiple of binomial coefficients and its
application, Amer. Math. Monthly 116 (2009), 836–839.
[2] V.J.W. Guo and J. Zeng, Some arithmetic properties of the q-Euler numbers and q-Salié
numbers, European J. Combin. 27 (2006), 884–895.
[3] D. Hanson, On the product of primes, Canad. Math. Bull. 15 (1972), 33–37.
[4] G.H. Hardy and E.M. Wright, The Theory of Numbers, 5th Ed., Oxford University Press,
London, 1979.
[5] D. Knuth and H. Wilf, The power of a prime that divides a generalized binomial coefficient,
J. Reine Angew. Math. 396 (1989), 212–219.
[6] T. Nagell, Introduction to Number Theory, Wiley, New York, 1951.
[7] M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), 126–
129.